First-Order Inconsistencies Caused by Rogue Trajectories

A theoretical requirement of the Interaction by Exchange with the Conditional Mean (IECM) micromixing model is that the mean concentration field produced by it must be consistent with the mean concentration field produced by a traditional Lagrangian stochastic (LS) marked particle model. We examine the violation of this requirement that occurs in a coupled LS–IECM model when unrealistically high particle velocities occur. No successful strategy was found to mitigate the effects of these rogue trajectories. It is our hope that this work will provide renewed impetus for investigation into rogue trajectories and methods to eliminate them from LS models.     

Rotation Of Trajectories In Lagrangian Stochastic Models Of Turbulent Dispersion

We present a new measure for the rotation of Lagrangian trajectories in turbulence that simplifies and generalises that suggested by Wilson and Flesch ( Boundary-Layer Meteorol. 84, 411–426). The new measure is the cross product of the velocity and acceleration and is directly related to the area, rather than the angle, swept out by the velocity vector. It makes it possible to derive a simple but exact kinematic expression for the mean rotation of the velocity vector and to partition this expression into terms that are closed in terms of Eulerian velocity moments up to second order and unclosed terms. The unclosed terms arise from the interaction of the fluctuating part of the velocity and the rate of change of the fluctuating velocity.We examine the mean rotation of a class of Lagrangian stochastic models that are quadratic in velocity for Gaussian inhomogeneous turbulence. For some of these models, including that of Thomson ( J. Fluid Mech. 180, 113–153), the unclosed part of the mean rotation vanishes identically, while for other models it is non-zero. Thus the mean rotation criterion clearly separates the class of models into two sets, but still does not provide a criterion for choosing a single model.We also show that models for which = 0 are independent of whether the model is derived on the assumption that total Lagrangian velocity is Markovian or whether the fluctuating part is Markovian.     

Comparing Two Implementations of a Micromixing Model. Part II: Canopy Flow

The sequential particle micromixing model (SPMMM) is used to estimate concentration fluctuations in plumes dispersing into a canopy flow. SPMMM uses the familiar single-particle Lagrangian stochastic (LS) trajectory framework to pre-calculate the required conditional mean concentrations, which are then used by an interaction by exchange with the conditional mean (IECM) micromixing model to predict the higher-order fluctuations of the scalar concentration field. The predictions are compared with experimental wind-tunnel dispersion data for a neutrally stratified canopy flow, and with a previously reported implementation using simultaneous particle trajectories. The two implementations of the LS–IECM model are shown to be largely consistent with one another and are able to simulate dispersion in a canopy flow with fair to good accuracy.     

Modelling Small-Scale Drifting Snow with a Lagrangian Stochastic Model Based on Large-Eddy Simulations

Observations of drifting snow on small scales have shown that, in spite of nearly steady winds, the snow mass flux can strongly fluctuate in time and space. Most drifting snow models, however, are not able to describe drifting snow accurately over short time periods or on small spatial scales as they rely on mean flow fields and assume equilibrium saltation. In an attempt to gain understanding of the temporal and spatial variability of drifting snow on small scales, we propose to use a model combination of flow fields from large-eddy simulations (LES) and a Lagrangian stochastic model to calculate snow particle trajectories and so infer snow mass fluxes. Model results show that, if particle aerodynamic entrainment is driven by the shear stress retrieved from the LES, we can obtain a snow mass flux varying in space and time. The obtained fluctuating snow mass flux is qualitatively compared to field and wind-tunnel measurements. The comparison shows that the model results capture the intermittent behaviour of observed drifting snow mass flux yet differences between modelled turbulent structures and those likely to be found in the field complicate quantitative comparisons. Results of a model experiment show that the surface shear-stress distribution and its influence on aerodynamic entrainment appear to be key factors in explaining the intermittency of drifting snow.     

A Lagrangian Stochastic Model for Sea-Spray Evaporation in the Atmospheric Marine Boundary Layer

The dispersion of heavy particles subjected to a turbulent forcing is often simulated with Lagrangian stochastic models. Although these models have been employed successfully over land, the implementation of traditional LS models in the marine boundary layer is significantly more challenging. We present an adaptation of traditional Lagrangian stochastic models to the atmospheric marine boundary layer with a particular focus on the representation of the scalar turbulence for temperature and humidity. In this new model, the atmosphere can be stratified and the bottom boundary is represented by a realistic wavy surface that moves and deforms. Hence, the correlation function for the turbulent flow following a particle is extended to the inhomogenous, anisotropic case. The results reproduce behaviour for scalar Lagrangian turbulence in a stratified airflow that departs only slightly from the expected behaviour in isotropic turbulence. When solving for the surface temperature and the radius of evaporating heavy water droplets in the airflow, the modelled turbulent forcing on the particle also behaves remarkably well. We anticipate that this model will prove especially useful in the context of sea-spray dispersion and its associated sensible heat, latent heat, and gas fluxes between spray droplets and the atmosphere.     

Comments on the 'Universality of the Lagrangian Velocity Structure Function Constant (C0) ACROSS DIFFERENT KINDS OF TURBULENCE’

Recently Du ( Boundary-Layer Meteorology 83, 207–219, 1997) estimated the value of the Lagrangian velocity structure constant, C0, in the inertial subrange by comparing experimental diffusion data and simulation results obtained with the one-dimensional form of Thomson's model ( J. Fluid Mech. 180, 529–556, 1987). Du reported that for several different flows (grid turbulence, a wind-tunnel boundary layer and the atmospheric surface layer under neutral stratification) the value of C0 is 3.0±0.5. Here, it is shown that optimal model agreement with experimental diffusion data for the wind-tunnel boundary layer is, in fact, obtained when C0=5.0 ± 0.5. It is also shown that accounting for the skewness of velocity statistics and finite Reynolds number effects does not significantly change this estimate for the value of C0. It is suggested that one-dimensional Lagrangian stochastic models are inconsistent with the supposed universality of C0.     

The footprint for flux measurements, from backward Lagrangian stochastic models

Backward Lagrangian stochastic models calculate particle trajectories in the atmosphere upstream of an observation point. A feature of these models is the ease with which the vertical flux contribution from surface area sources can be calculated. The flux contribution at an observation point P is found by summing the ratio of the particle vertical velocity at P to the touchdown velocity for particles which impact the ground within the source boundary.     

A Three-Dimensional Backward Lagrangian Footprint Model For A Wide Range Of Boundary-Layer Stratifications

We present a three-dimensional Lagrangian footprint model with the ability to predict the area of influence (footprint) of a measurement within a wide range of boundary-layer stratifications and receptor heights. The model approach uses stochastic backward trajectories of particles and satisfies the well-mixed condition in inhomogeneous turbulence for continuous transitions from stable to convective stratification. We introduce a spin-up procedure of the model and a statistical treatment of particle touchdowns which leads to a significant reduction of CPU time compared to conventional footprint modelling approaches. A comparison with other footprint models (of the analytical and Lagrangian type) suggests that the present backward Lagrangian model provides valid footprint predictions under any stratification and, moreover, for applications that reach across different similarity scaling domains (e.g., surface layer to mixed layer, for use in connection with aircraft measurements or with observations on high towers).     

A Lagrangian stochastic model for the trajectories of particle pairs and its application to the prediction of concentration variance within plant canopies

A simple Lagrangian stochastic model for the trajectories of particle pairs in high Reynolds-number turbulent flows is presented. In this model, the velocities of particle pairs are initially correlated but subsequently each particle moves independently. The independent single-particle trajectories are simulated using Thomson's model [J. Fluid Mech. 180, 529–556, 1987]. This two-particle model exactly satisfies the well-mixed condition for Gaussian turbulence when length scales, characterizing the two-point Eulerian velocity correlation function, vanish. Temperature variances, due to heat released as a passive scalar from an elevated plane source, within a model plant canopy (Coppin et al. Boundary Layer Meteorol. 35, 167–191, 1986) are shown to be well predicted by the model. It is suggested that for strongly inhomogeneous flows, the two-point Eulerian velocity function is of secondary importance in determining the simulated trajectories of particle pairs compared to the importance of ensuring satisfaction of the two-to-one constraint (Borgas and Sawford. J. Fluid Mech. 279, 69–99, 1994); i.e ensuring that one-particle statistics obtained from the two-particle model are the same as those obtained from the corresponding one-particle model. Limitations of this modelling approach are discussed.     

Instability in Lagrangian stochastic trajectory models,and a method for its cure

A number of authors have reported the problem of unrealistic velocities (“rogue trajectories”) when computing the paths of particles in a turbulent flow using modern Lagrangian stochastic (LS) models, and have resorted to ad hoc interventions. We suggest that this problem stems from two causes: (1) unstable modes that are intrinsic to the dynamical system constituted by the generalized Langevin equations, and whose actual triggering (expression) is conditional on the fields of the mean velocity and Reynolds stress tensor and is liable to occur in complex, disturbed flows (which, if computational, will also be imperfect and discontinuous); and, (2) the “stiffness” of the generalized Langevin equations, which implies that the simple stochastic generalization of the Euler scheme usually used to integrate these equations is not sufficient to keep round-off errors under control. These two causes are connected, with the first cause (dynamical instability) exacerbating the second (numerical instability); removing the first cause does not necessarily correct the second, and vice versa. To overcome this problem, we introduce a fractional-step integration scheme that splits the velocity increment into contributions that are linear (U _i ) and nonlinear (U _i U _j ) in the Lagrangian velocity fluctuation vector U, the nonlinear contribution being further split into its diagonal and off-diagonal parts. The linear contribution and the diagonal part of the nonlinear contribution to the solution are computed exactly (analytically) over a finite timestep Δt, allowing any dynamical instabilities in the system to be diagnosed and removed, and circumventing the numerical instability that can potentially result in integrating stiff equations using the commonly applied explicit Euler scheme. We contrast results using this and the primitive Euler integration scheme for computed trajectories in a drastically inhomogeneous urban canopy flow.     

Lagrangian Stochastic Models For Turbulent Dispersion In The Atmospheric Boundary Layer

A one-particle three-dimensional stochastic Lagrangian model fortransport of particles in a horizontally-homogeneous atmosphericsurface layer with arbitrary one-point probability density functionof Eulerian velocity fluctuations is suggested. A uniquely definedLagrangian stochastic model in the class of well-mixed models isconstructed from physically plausible assumptions. These assumptionsare: (i) in the neutrally stratified horizontally homogeneous surface layer, the vertical motion is mainly controlled by eddies whose size is of order of the current height; and (ii), the streamwise drift term is independent of the crosswind velocity. Numerical simulations for neutral stratification have shown a good agreement of our model with the well-known Thomson's model, with Flesch and Wilson's model, and with experimental measurements as well. However there is a discrepancy of these results with the results obtained by Reynolds' model.     

A Homogeneous Langevin Equation Model, Part i: Simulation of Particle Trajectories in Turbulence with a Skewed Velocity Distribution

A Lagrangian stochastic model for the time evolution of the velocity of a fluid particle is presented. This model is based on a one-dimensional generalized Langevin equation, and assumes the velocity probability distribution of the turbulent fluid is skewed and spatially homogeneous. This has been shown to be an effective approach to simulating vertical dispersion in the convective boundary layer. We use a form of the Langevin equation that has a linear (in velocity) deterministic acceleration and a random acceleration that is a non-Gaussian, skewed process. For the case of homogeneous fluid velocity statistics, this 'linear-skewed' Langevin equation can be integrated explicitly, resulting in an efficient numerical simulation method. Model simulations were tested using cases for which exact, analytic statistical properties of particle velocity are known. Results of these tests show that, for homogeneous turbulence, a linear-skewed Langevin equation model can overcome the difficulties encountered in applying a Langevin equation with a skewed random acceleration. The linear-skewed Langevin equation model results are compared to results of a 'nonlinear-Gaussian' Langevin equation model, and show that the linear-skewed model is significantly more efficient.     

On the Formulation of Lagrangian Stochastic Models of Scalar Dispersion Within Plant Canopies

Lagrangian stochastic models, quadratic in velocity and satisfying the well-mixed condition for two-dimensional Gaussian turbulence, are used to make predictions of scalar dispersion within a model plant canopy. The non-uniqueness associated with satisfaction of the well-mixed condition is shown to be non-trivial (i.e. different models produce different predictions for scalar dispersion). The best agreement between measured and predicted mean concentrations of scalars is shown to be obtained with a small sub-class of optimal models. This sub-class of optimal models includes Thomson's model (J. Fluid Mech. 180, 529–556, 1987), the simplest model that satisfies the well-mixed condition for Gaussian turbulence, but does not include two other models identified recently as being in optimal agreement with the measured spread of tracers in a neutral boundary layer. It is therefore demonstrated that such models are not universal, i.e. applicable to a wide range of flows without readjustment of model parameters. Predictions for scalar dispersion in the model plant canopy are also obtained using the model of Flesch and Wilson (Boundary-Layer Meteorol. 61, 349–374, 1992). It is shown that, when used with a Gaussian velocity distribution or a maximum-missing-information velocity distribution, which accounts for the measured skewness and kurtosis of velocity statistics, the agreement between predictions obtained using the model of Flesch and Wilson and measurements is as good as that obtained using Thomson's model.     

Numerical Considerations for Lagrangian Stochastic Dispersion Models: Eliminating Rogue Trajectories,and the Importance of Numerical Accuracy

When Lagrangian stochastic models for turbulent dispersion are applied to complex atmospheric flows, some type of ad hoc intervention is almost always necessary to eliminate unphysical behaviour in the numerical solution. Here we discuss numerical strategies for solving the non-linear Langevin-based particle velocity evolution equation that eliminate such unphysical behaviour in both Reynolds-averaged and large-eddy simulation applications. Extremely large or ‘rogue’ particle velocities are caused when the numerical integration scheme becomes unstable. Such instabilities can be eliminated by using a sufficiently small integration timestep, or in cases where the required timestep is unrealistically small, an unconditionally stable implicit integration scheme can be used. When the generalized anisotropic turbulence model is used, it is critical that the input velocity covariance tensor be realizable, otherwise unphysical behaviour can become problematic regardless of the integration scheme or size of the timestep. A method is presented to ensure realizability, and thus eliminate such behaviour. It was also found that the numerical accuracy of the integration scheme determined the degree to which the second law of thermodynamics or ‘well-mixed condition’ was satisfied. Perhaps more importantly, it also determined the degree to which modelled Eulerian particle velocity statistics matched the specified Eulerian distributions (which is the ultimate goal of the numerical solution). It is recommended that future models be verified by not only checking the well-mixed condition, but perhaps more importantly by checking that computed Eulerian statistics match the Eulerian statistics specified as inputs.     

Three-Dimensional Lagrangian Simulation of Suspended Particles in the Neutrally Stratified Atmospheric Surface Layer

A three-dimensional (3-D) inertial particle – Lagrangian stochastic model for heavy particles in turbulent flows has been constructed. In this model, particle velocities are computed by adopting a non-linear drag law, while fluid velocity in the vicinity of a particle is calculated using a 3-D Langevin equation. Our model results have shown that the inclusion of the horizontal fluid velocity fluctuation computations and a non-linear drag law have an impact on the statistics of both fluid and particles when compared with our earlier one-dimensional (1-D) model with a linear drag law. Model results are compared and contrasted with Businger’s 1965 theory in terms of effective settling velocity.     

Lagrangian Stochastic Modeling of Dispersion in the Stable Boundary Layer

Lagrangian stochastic models are well-suited for modeling dispersion in the stable boundary layer, especially in complex terrain. This note briefly describes the formulations and application of a Lagrangian stochastic model to predict dispersion of tracers released within nocturnal drainage flows.     

The Volumetric Particle Approach for Concentration Fluctuations and Chemical Reactions in Lagrangian Particle and Particle-grid Models

A new approach is proposed to predict concentration fluctuations in the framework of one-particle Lagrangian stochastic models. The approach is innovative since it allows the computation of concentration fluctuations in dispersing plumes using a Lagrangian one-particle model with micromixing but with no need for the simulating of background particles. The extension of the model for the treatment of chemically reactive plumes is also accomplished and allows the computation of plume-related chemical reactions in a Lagrangian one-particle framework separately from the background chemical reactions, accounting for the effect of concentration fluctuations on chemical reactions in a general, albeit approximate, manner. These characteristics should make the proposed approach an ideal tool for plume-in-grid calculations in chemistry transport models. The results are compared to the wind-tunnel experiments of Fackrell and Robins (J Fluid Mech, 117:1–26, 1982) for plume dispersion in a neutral boundary layer and to the measurements of Legg et al. (Boundary-Layer Meteorol, 35:277–302, 1986) for line source dispersion in and above a model canopy. Preliminary reacting plume simulations are also shown comparing the model with the experimental results of Brown and Bilger (J Fluid Mech, 312:373–407, 1996; Atmos Environ, 32:611–628, 1998) to demonstrate the feasibility of computing chemical reactions in the proposed framework.     

The Canadian Meteorological Centre's Atmospheric Transport and Dispersion Modelling Suite

This paper describes the integrated suite of Lagrangian transport and dispersion models in operation at the Canadian Meteorological Centre. These models have been in use for several years and are applied to many types of environmental emergencies covering spatial scales from the very local to the global. The Modèle Lagrangien Courte Distance (MLCD) is used for atmospheric spills of the order of a few kilometres. The Modèle Lagrangien de dispersion de particules d'ordre 1 (MLDP1) is normally used for events affecting areas less than 100?km; Modèle Lagrangien dispersion de particules d'ordre zéro (MLDP0) is used for events of continental and global consequences. The Modèle Lagrangien dispersion de particules mode mixte (MLDPmm) alternates between first-order and zeroth-order depending on criteria specified by the user. The theoretical bases of the models are presented, and the main algorithms used in their implementation are discussed. Modelling of the diffusion processes is based on a stochastic differential equation with the assumption of quasi-stationary Gaussian turbulence, locally homogeneous in the horizontal. The practical aspects of the operational implementation are also described. Using these models, results from simulations of real cases on scales ranging from the very local, to a few kilometres, to regional (approximately 100?km) to continental (approximately 1000?km) and to global (approximately 10,000?km) are compared and validated with available observational data.     

Comparison of the Langrangian Footprint

We compare flux and concentration footprint estimates of athree-dimensional Lagrangian stochastic dispersion modelapplying backward trajectories with the results of ananalytical footprint model by Kormann and Meixner.The comparison is performed for varying stability regimesof the surface layer as well as for different measurementheights. In general, excellent correspondence is found.